most recent 30 from http://mathoverflow.net 2013-05-25T08:17:52Z http://mathoverflow.net/feeds/question/57851 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57851/are-all-shimura-varieties-special-subvarieties-of-the-siegel-modular-variety jacob 2011-03-08T16:31:03Z 2011-03-08T21:17:47Z

<p>Given a Shimura variety $S$, is it possible to imbed $S$ as a special Subvariety of the Siegel modular variety $A_{g,N}$, for some $g$ and level $N$? I expect that the answer is yes, essentially since every semisimple group over $\mathbb{Q}$ should imbed into $GL_n$ via its adjoint representation, and $GL_n$ imbeds into $SP_{2n}$. However, I'm a bit worried about the business regarding weights.</p> <p>Thank you, Jacob</p>

http://mathoverflow.net/questions/57851/are-all-shimura-varieties-special-subvarieties-of-the-siegel-modular-variety/57883#57883 Keerthi Madapusi Pera 2011-03-08T21:17:47Z 2011-03-08T21:17:47Z

<p>The answer is no, in general. The problem is to find an embedding so that the minuscule character corresponding to the Shimura datum for $S$ induces the minuscule character of $GSp_{2n}$ corresponding to a decomposition into Lagrangians. </p> <p>In the affirmative direction, for most classical, simply connected groups (and only for classical groups, i.e. of types $A$,$B$,$C$ and $D$), the answer is yes; some subtleties crop up for $Spin^*(2n)$ (this is the so called $D^{\mathbb{H}}$ case), for which only the quotient by an order 2 central sub-group admits a symplectic embedding (of Shimura data).</p> <p>This is all beautifully laid out in Deligne's article 'Varietes de Shimura...' <a href="http://www.ams.org/mathscinet-getitem?mr=0546620" rel="nofollow">here</a>, following Satake <a href="http://www.jstor.org/stable/2373012" rel="nofollow">here</a>. See also Proposition 1.21 in Milne's article <a href="http://www.jmilne.org/math/articles/1994bP.pdf" rel="nofollow">here</a> </p>